let X be Complex_Banach_Algebra; :: thesis: for x, y, z being Element of X
for a, b being Complex holds
( a * (x * y) = (a * x) * y & a * (x + y) = (a * x) + (a * y) & (a + b) * x = (a * x) + (b * x) & (a * b) * x = a * (b * x) & (a * b) * (x * y) = (a * x) * (b * y) & a * (x * y) = x * (a * y) & (0. X) * x = 0. X & x * (0. X) = 0. X & x * (y - z) = (x * y) - (x * z) & (y - z) * x = (y * x) - (z * x) & (x + y) - z = x + (y - z) & (x - y) + z = x - (y - z) & (x - y) - z = x - (y + z) & x + y = (x - z) + (z + y) & x - y = (x - z) + (z - y) & x = (x - y) + y & x = y - (y - x) & ( ||.x.|| = 0 implies x = 0. X ) & ( x = 0. X implies ||.x.|| = 0 ) & ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(x * y).|| <= ||.x.|| * ||.y.|| & ||.(1. X).|| = 1 )
let x, y, z be Element of X; :: thesis: for a, b being Complex holds
( a * (x * y) = (a * x) * y & a * (x + y) = (a * x) + (a * y) & (a + b) * x = (a * x) + (b * x) & (a * b) * x = a * (b * x) & (a * b) * (x * y) = (a * x) * (b * y) & a * (x * y) = x * (a * y) & (0. X) * x = 0. X & x * (0. X) = 0. X & x * (y - z) = (x * y) - (x * z) & (y - z) * x = (y * x) - (z * x) & (x + y) - z = x + (y - z) & (x - y) + z = x - (y - z) & (x - y) - z = x - (y + z) & x + y = (x - z) + (z + y) & x - y = (x - z) + (z - y) & x = (x - y) + y & x = y - (y - x) & ( ||.x.|| = 0 implies x = 0. X ) & ( x = 0. X implies ||.x.|| = 0 ) & ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(x * y).|| <= ||.x.|| * ||.y.|| & ||.(1. X).|| = 1 )
let a, b be Complex; :: thesis: ( a * (x * y) = (a * x) * y & a * (x + y) = (a * x) + (a * y) & (a + b) * x = (a * x) + (b * x) & (a * b) * x = a * (b * x) & (a * b) * (x * y) = (a * x) * (b * y) & a * (x * y) = x * (a * y) & (0. X) * x = 0. X & x * (0. X) = 0. X & x * (y - z) = (x * y) - (x * z) & (y - z) * x = (y * x) - (z * x) & (x + y) - z = x + (y - z) & (x - y) + z = x - (y - z) & (x - y) - z = x - (y + z) & x + y = (x - z) + (z + y) & x - y = (x - z) + (z - y) & x = (x - y) + y & x = y - (y - x) & ( ||.x.|| = 0 implies x = 0. X ) & ( x = 0. X implies ||.x.|| = 0 ) & ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(x * y).|| <= ||.x.|| * ||.y.|| & ||.(1. X).|| = 1 )
thus ( a * (x * y) = (a * x) * y & a * (x + y) = (a * x) + (a * y) & (a + b) * x = (a * x) + (b * x) & (a * b) * x = a * (b * x) ) by CFUNCDOM:def 9, CLVECT_1:def 2, CLVECT_1:def 3, CLVECT_1:def 4; :: thesis: ( (a * b) * (x * y) = (a * x) * (b * y) & a * (x * y) = x * (a * y) & (0. X) * x = 0. X & x * (0. X) = 0. X & x * (y - z) = (x * y) - (x * z) & (y - z) * x = (y * x) - (z * x) & (x + y) - z = x + (y - z) & (x - y) + z = x - (y - z) & (x - y) - z = x - (y + z) & x + y = (x - z) + (z + y) & x - y = (x - z) + (z - y) & x = (x - y) + y & x = y - (y - x) & ( ||.x.|| = 0 implies x = 0. X ) & ( x = 0. X implies ||.x.|| = 0 ) & ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(x * y).|| <= ||.x.|| * ||.y.|| & ||.(1. X).|| = 1 )
thus (a * b) * (x * y) = a * (b * (x * y)) by CLVECT_1:def 4
.= a * (x * (b * y)) by CLOPBAN2:def 11
.= (a * x) * (b * y) by CFUNCDOM:def 9 ; :: thesis: ( a * (x * y) = x * (a * y) & (0. X) * x = 0. X & x * (0. X) = 0. X & x * (y - z) = (x * y) - (x * z) & (y - z) * x = (y * x) - (z * x) & (x + y) - z = x + (y - z) & (x - y) + z = x - (y - z) & (x - y) - z = x - (y + z) & x + y = (x - z) + (z + y) & x - y = (x - z) + (z - y) & x = (x - y) + y & x = y - (y - x) & ( ||.x.|| = 0 implies x = 0. X ) & ( x = 0. X implies ||.x.|| = 0 ) & ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(x * y).|| <= ||.x.|| * ||.y.|| & ||.(1. X).|| = 1 )
thus a * (x * y) = x * (a * y) by CLOPBAN2:def 11; :: thesis: ( (0. X) * x = 0. X & x * (0. X) = 0. X & x * (y - z) = (x * y) - (x * z) & (y - z) * x = (y * x) - (z * x) & (x + y) - z = x + (y - z) & (x - y) + z = x - (y - z) & (x - y) - z = x - (y + z) & x + y = (x - z) + (z + y) & x - y = (x - z) + (z - y) & x = (x - y) + y & x = y - (y - x) & ( ||.x.|| = 0 implies x = 0. X ) & ( x = 0. X implies ||.x.|| = 0 ) & ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(x * y).|| <= ||.x.|| * ||.y.|| & ||.(1. X).|| = 1 )
A1: (x * (y - z)) + (x * z) = x * ((y - z) + z) by VECTSP_1:def 2
.= x * (y - (z - z)) by RLVECT_1:29
.= x * (y - (0. X)) by RLVECT_1:15
.= x * y by RLVECT_1:13 ;
x * (0. X) = x * ((0. X) + (0. X)) by RLVECT_1:def 4
.= (x * (0. X)) + (x * (0. X)) by VECTSP_1:def 2 ;
then 0. X = ((x * (0. X)) + (x * (0. X))) - (x * (0. X)) by RLVECT_1:15;
then 0. X = (x * (0. X)) + ((x * (0. X)) - (x * (0. X))) by RLVECT_1:def 3;
then A2: 0. X = (x * (0. X)) + (0. X) by RLVECT_1:15;
(0. X) * x = ((0. X) + (0. X)) * x by RLVECT_1:def 4
.= ((0. X) * x) + ((0. X) * x) by VECTSP_1:def 3 ;
then 0. X = (((0. X) * x) + ((0. X) * x)) - ((0. X) * x) by RLVECT_1:15;
then 0. X = ((0. X) * x) + (((0. X) * x) - ((0. X) * x)) by RLVECT_1:def 3;
then 0. X = ((0. X) * x) + (0. X) by RLVECT_1:15;
hence ( (0. X) * x = 0. X & x * (0. X) = 0. X ) by A2, RLVECT_1:def 4; :: thesis: ( x * (y - z) = (x * y) - (x * z) & (y - z) * x = (y * x) - (z * x) & (x + y) - z = x + (y - z) & (x - y) + z = x - (y - z) & (x - y) - z = x - (y + z) & x + y = (x - z) + (z + y) & x - y = (x - z) + (z - y) & x = (x - y) + y & x = y - (y - x) & ( ||.x.|| = 0 implies x = 0. X ) & ( x = 0. X implies ||.x.|| = 0 ) & ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(x * y).|| <= ||.x.|| * ||.y.|| & ||.(1. X).|| = 1 )
thus x * (y - z) = (x * (y - z)) + (0. X) by RLVECT_1:4
.= (x * (y - z)) + ((x * z) - (x * z)) by RLVECT_1:15
.= (x * y) - (x * z) by A1, RLVECT_1:def 3 ; :: thesis: ( (y - z) * x = (y * x) - (z * x) & (x + y) - z = x + (y - z) & (x - y) + z = x - (y - z) & (x - y) - z = x - (y + z) & x + y = (x - z) + (z + y) & x - y = (x - z) + (z - y) & x = (x - y) + y & x = y - (y - x) & ( ||.x.|| = 0 implies x = 0. X ) & ( x = 0. X implies ||.x.|| = 0 ) & ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(x * y).|| <= ||.x.|| * ||.y.|| & ||.(1. X).|| = 1 )
A3: ((y - z) * x) + (z * x) = ((y - z) + z) * x by VECTSP_1:def 3
.= (y - (z - z)) * x by RLVECT_1:29
.= (y - (0. X)) * x by RLVECT_1:15
.= y * x by RLVECT_1:13 ;
thus (y - z) * x = ((y - z) * x) + (0. X) by RLVECT_1:4
.= ((y - z) * x) + ((z * x) - (z * x)) by RLVECT_1:15
.= (y * x) - (z * x) by A3, RLVECT_1:def 3 ; :: thesis: ( (x + y) - z = x + (y - z) & (x - y) + z = x - (y - z) & (x - y) - z = x - (y + z) & x + y = (x - z) + (z + y) & x - y = (x - z) + (z - y) & x = (x - y) + y & x = y - (y - x) & ( ||.x.|| = 0 implies x = 0. X ) & ( x = 0. X implies ||.x.|| = 0 ) & ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(x * y).|| <= ||.x.|| * ||.y.|| & ||.(1. X).|| = 1 )
thus (x + y) - z = x + (y - z) by RLVECT_1:def 3; :: thesis: ( (x - y) + z = x - (y - z) & (x - y) - z = x - (y + z) & x + y = (x - z) + (z + y) & x - y = (x - z) + (z - y) & x = (x - y) + y & x = y - (y - x) & ( ||.x.|| = 0 implies x = 0. X ) & ( x = 0. X implies ||.x.|| = 0 ) & ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(x * y).|| <= ||.x.|| * ||.y.|| & ||.(1. X).|| = 1 )
thus (x - y) + z = x - (y - z) by RLVECT_1:29; :: thesis: ( (x - y) - z = x - (y + z) & x + y = (x - z) + (z + y) & x - y = (x - z) + (z - y) & x = (x - y) + y & x = y - (y - x) & ( ||.x.|| = 0 implies x = 0. X ) & ( x = 0. X implies ||.x.|| = 0 ) & ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(x * y).|| <= ||.x.|| * ||.y.|| & ||.(1. X).|| = 1 )
thus (x - y) - z = x - (y + z) by RLVECT_1:27; :: thesis: ( x + y = (x - z) + (z + y) & x - y = (x - z) + (z - y) & x = (x - y) + y & x = y - (y - x) & ( ||.x.|| = 0 implies x = 0. X ) & ( x = 0. X implies ||.x.|| = 0 ) & ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(x * y).|| <= ||.x.|| * ||.y.|| & ||.(1. X).|| = 1 )
thus (x - z) + (z + y) = ((x - z) + z) + y by RLVECT_1:def 3
.= (x - (z - z)) + y by RLVECT_1:29
.= (x - (0. X)) + y by RLVECT_1:15
.= x + y by RLVECT_1:13 ; :: thesis: ( x - y = (x - z) + (z - y) & x = (x - y) + y & x = y - (y - x) & ( ||.x.|| = 0 implies x = 0. X ) & ( x = 0. X implies ||.x.|| = 0 ) & ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(x * y).|| <= ||.x.|| * ||.y.|| & ||.(1. X).|| = 1 )
thus (x - z) + (z - y) = ((x - z) + z) - y by RLVECT_1:def 3
.= (x - (z - z)) - y by RLVECT_1:29
.= (x - (0. X)) - y by RLVECT_1:15
.= x - y by RLVECT_1:13 ; :: thesis: ( x = (x - y) + y & x = y - (y - x) & ( ||.x.|| = 0 implies x = 0. X ) & ( x = 0. X implies ||.x.|| = 0 ) & ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(x * y).|| <= ||.x.|| * ||.y.|| & ||.(1. X).|| = 1 )
thus (x - y) + y = x - (y - y) by RLVECT_1:29
.= x - (0. X) by RLVECT_1:15
.= x by RLVECT_1:13 ; :: thesis: ( x = y - (y - x) & ( ||.x.|| = 0 implies x = 0. X ) & ( x = 0. X implies ||.x.|| = 0 ) & ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(x * y).|| <= ||.x.|| * ||.y.|| & ||.(1. X).|| = 1 )
thus y - (y - x) = (y - y) + x by RLVECT_1:29
.= (0. X) + x by RLVECT_1:15
.= x by RLVECT_1:4 ; :: thesis: ( ( ||.x.|| = 0 implies x = 0. X ) & ( x = 0. X implies ||.x.|| = 0 ) & ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(x * y).|| <= ||.x.|| * ||.y.|| & ||.(1. X).|| = 1 )
thus ( ( ||.x.|| = 0 implies x = 0. X ) & ( x = 0. X implies ||.x.|| = 0 ) & ||.(a * x).|| = |.a.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(x * y).|| <= ||.x.|| * ||.y.|| ) by CLOPBAN2:def 9, CLVECT_1:def 13, NORMSP_0:def 5; :: thesis: ||.(1. X).|| = 1
thus ||.(1. X).|| = 1 by CLOPBAN2:def 10; :: thesis: verum